However, in the linear elastic case, we have a linear proportion between the cauchy stress tensor and the small deformation strain tensor. Combining the above two conditions and knowing the relation, by the use of. Such internal forces are typical, for instance, for the motion of heliotropic plants, and for muscle tissue. We do not derive this functional here, but rather see what are the implications of this form. Difference between first and second piolakirchhoff stress vector. The first piola kirchhoff stress tensor, a eulerianlagrangian twopoint tensor, is also obtained directly from the second piola kirchhoff stress tensor. Our aim is to model and study the cohesion forces or internal forces of a system, that is to say the actions exerted by part of a system s on another part of s. Ii stress 1 stress tensors pierre badel emse ds 0 ds n n df df df. Everything here applies regardless of the type of stress tensor. In a general model of continuous media, where it is not assumed that the stress tensor is symmetric, this means that we need 9 scalar quantities to characterize the state of. The transform applies to any stress tensor, or strain tensor for that matter. Stress tensor symbol areaweighted normal force relations cauchy.
Stress used in lagrangian equation for solid mechanics. Piolakirchhoff stress tensor according to the linearized finite theory of elasticity for hyperelastic. A key discriminator among the different stress tensors is whether they report stress in a materials undeformed, and especially unrotated state, the reference configuration, or in its deformed and rotated state, the current configuration. In tensor notation, the state of stress at a point is expressed as where i and j are iterated over x, y, and z.
Note that unlike the cauchy stress tensor t, the piolakirchhoff stress tensor t. In the case of finite deformations, the piola kirchhoff stress tensors express the stress relative to the reference configuration. Recrystallization and texture evolution during hot rolling of. An inner pressure or an inner fiber tension is acting as a driving force. Why the second piolakirchhoff stress is the derivative of. Since the biot stress tensor tb is not positive definite this decomposition is not unique, in general. Nov 21, 20 the stress is defined as the force in the current configuration acting on the original area. The piolakirchhoff stress may depend linearly on the. The workconjugate objective stress rates, giving the correct secondorder incremental work, are the truesdell rate for m2, biot rate for 1, jaumann rate of kirchhoff stress for m0, and lie derivative of kirchhoff stress. The cauchy stress tensor and the piola kirchhoff tensor. Massspring model for simulation of heart valve tissue.
Automatic generation and discretization of fully periodic. The sas shell formulation utilizes the objective greenlagrange strain tensor that exactly represents the arbitrarily large rigidbody motions of a shell in any curvilinear coordinate system. Introduction classical elasticity is a phenomenological theory to describe the properties of solids. Four possibilities for stress tensors and their relationship to the cauchy stress. Tractionfree internal and external lateral surfaces have been considered as suitable boundary conditions, while perfect contact with rigid surfaces has been modeled for the upper and lower faces. A material body b fxgis a compact measurable set of an in nite number of material elements x, called the material particles or material points, that can be placed in a onetoone correspondence with triplets of real numbers. Our study leads to the definition of the cauchy stress tensor and to the equations of statics and dynamics that then follow by application of the fundamental law of dynamics.
Notice that p is a twopoint tensor possessing the geometrical mapping. Samer adeeb first and second piolakirchhoff stress tensors. Note that detf appears in the elastic energy functional due to the incompressibility condition. The 2nd piola kirchhoff stress tensor and the greenlagrange strain tensor are energetically conjugate. Optimal control problems in finitestrain elasticity by inner. Nonstandard objective functions relevant for elasticity problems are introduced. Kyuichiro washizu on the occasion of his 60th birthday and the completion of a distinguished academic career at the university of tokyo. The stress measure could be any of the stress measures discussed earlier, such as the cauchy stress tensor, the second piola kirchhoff stress tensor.
The static spherically symmetric body in relativistic elasticity. A jintegralbased arclength solver for brittle and ductile. Geometrically speaking aj isabundle mapoftm into tnoverj. May 18, 2011 a novel computational approach for the dynamic analysis of a large scale rigidflexible multibody system composed of composite laminated plates is proposed. Two often used for large deformation mechanics are the 1st piola kirchoff pk stress tensor and the 2nd piola kirchoff pk stress tensor. The subject of all studies in continuum mechanics, and the domain of all physical quantities, is the material body. The state of stress at a given point of the earths crust, at a given moment, is represented by a second order tensor.
Hi naveen, yes, the atc package can compute hardy stress for an eam type potential. The cauchy stress tensor is expressed in the eulerian variable. This permits one to calculate the transverse components of the second piola kirchhoff stress tensor in piezoelectric doublycurved shells with a high. If the stress tensor is symmetric such as the cauchy stress and the second piola kirchhoff stress, the singular value decomposition becomes the same as the eigenvalue decomposition, and the principal values coincide with the eigenvalues of the stress tensor. Topic 3 lagrangian continuum mechanics variables for analysis. Kirchhoff stress in nonlinear thick and thin structures using exact geometry sas solid. Tokeepthefullgenerality weassurnethat adepends onthe actual configuration j,possibly inanonlocalway. N ds df s n ds i ij j s piola kirchhoff 2 stress tensor pk2. Conservation laws and mechanical equilibria continuum. The most commonly used measure of stress is the cauchy stress tensor, often called simply the stress tensor or true stress. We point out that although here, for convenience, we consider an incompressible hyperelastic material, our approach can be applied to study other types of materials as well. It is shown that, whenever the residual stress does not vanish, the response function delivering the piola kirchhoff stress in terms of the deformation gradient may be genuinely linear, and yet independent of the observer.
As you know, in the structural mechanics module, when i insert the strain energy function and i activate the equation view, i can already find a definition of the first piola kirchhoff tensor. The stress tensor the secondorder tensor which we will be examining has. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity. The kirchhoff stress is just the cauchy stress scaled by the volume change. The need for materials with applicationspecific properties motivates research on fiberreinforced composites. Chapter 3 the stress tensor for a fluid and the navier stokes equations 3. Optimal control problems for finitestrain elasticity are considered. Second piola kirchhoff stress tensor the second piola stress tensor was concocted to be a symmetric tensor. Both the first piola kirchhoff stress tensor and the nominal stress tensor are widely used in. Assessment of second piolakirchhoff stress tensor in. Introduction this page introduces several definitions of stress.
While the balance of linear momentum and the balance of energy are stated in weak form, we retain the kinematic relationship 31 in local form. The second piola kirchhoff stress or pk2 stress s \displaystyle \boldsymbol s. The second piolakirchhoff stress tensor, is related to the second piolakirchhoff stress tensor in the intermediate configuration through, and the cauchy stress tensor, in each grain is given by. The first piola kirchhoff stress tensor, an eulerianlagrangian twopoint tensor, is also obtained directly from the second piola kirchhoff stress tensor. Two often used for large deformation mechanics are the 1st piolakirchoff pk stress tensor and the 2nd piolakirchoff pk stress tensor. Optimality conditions are derived on a formal basis, and a. The red vectors in figure 1 show a schematic of the forces acting on the deformed area vectors and when viewed in the deformed configuration or when viewed in the undeformed configuration. Difference between first and second piolakirchhoff stress. Moreover, in the absence of external force densities, we will investigate the virtual work depending on any configuration and any infinitesimal virtual distortion. This is sometimes useful in doing computations for instance using the finite element method for large deformation problems. Dec 09, 2015 made with explain everything for continuum mechanics.
Corotated cauchy stress tensor su this symmetric stress tensor was introduced by. Therefore, we need to develop alternative stress tensors. Almansi lagrangian strain tensor, and tends to 0 for the logarithmic hencky strain tensor 1,2. While the balance of linear momentum and the balance of energy are stated in weak form, we retain the kinematic relationship 3 1 in local form. Analysis of some basic approaches to finite strain elastoplasticity in. We discuss the behavior of the systems for large deformations. Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are. The term nominal stress tensor is sometimes used in the literature in reference to the first piolakirchhoff stress or its transpose. Piola kirchhoff stress tensor by the reference cauchy theorem t. Since the 1st piola kirchhoff stress tensor is notsymmetric, one can create a symmetric tensor as.
Systolic contraction was modelled by defining the total second piolakirchhoff stress tensor s as the sum of a passive part, s passive, derived from the relation according to eq. Dynamics of a large scale rigidflexible multibody system. Chapter 3 the stress tensor for a fluid and the navier. Lectures for transport phenomena course at olin college this lecture describes what the stress tensor is. Oct, 2015 the answer to this question lies at the realm of intermingling of thermodynamics and nonlinear fem. Please take a look at examplesuseratchardy for examples that include both calculation of the 1st piola kirchhoff stress tensor e. Phase transitions in a slender cylinder composed of an. I am wondering where, when the structural mechanics module is used, i can see the definition of the first piola kirchhoff tensor. Electromagnetic stressenergy tensor, in relativistic physics. It is interesting that most, perhaps even all, stress definitions can be paired with.
Droplet motion driven by tensotaxis purdue university. It is used heavily in applications such as structural mechanics and other engineering disciplines. Both mathematical and engineering mi stakes are easily made if this crucial difference is not recognized and understood. The 1st piolakirchhoff stress is the 3d generalization of the 1d concept of engineering stress. Errors caused by nonwork conjugate stress and strain. Venant materials but differ in the strain variable used. This is not a symmetric tensor and for computational ease, this we use a the second piola kirchoff stress which is symmetric in nature as it is a result of the product between the finger tensor. For infinitesimal deformations and rotations, the cauchy and piola. The transpose of the first piola kirchhoff stress tensor is the socalled nominal stress tensor. The rigid parts in the system are described through the natural coordinate formulation ncf and the flexible bodies in the system are modeled via the finite elements of absolute nodal coordinate formulation ancf, which can lead to a.
If the material rotates without a change in stress state rigid rotation, the components of the 1st piolakirchhoff stress tensor will vary with material orientation. The first piola kirchoff stress tensor relates the cauchy stress tensor to the stress. Let t be the cauchy stress tensor also known as true stresses at a certain material point. This is in contrast to the cauchy stress tensor which expresses the stress relative to the present configuration. Effective stress tensor an overview sciencedirect topics. Relativistic elasticity nonlinear elasticity in general. The corresponding second piolakirchhoff stress is 45 s.
Prove that the second piolakirchhoff stress tensor is also symmetric. If the material rotates without a change in stress state rigid rotation, the components of the 1st piola kirchhoff stress tensor will vary with material orientation. The 1st piola kirchhoff stress is energy conjugate to the deformation gradient. What is the tensor type for green lagrange strain tensor and 2nd 2nd piola kirchhoff tensor.
The first piolakirchhoff stress tensor and the virtual work in this section we will present the characterization of a medium via a first piolakirchhoff stress tensor. The last section provided the first step toward finding the stress on an arbitrary plane in continuous media. For clarity in what follows we drop the superscript s and the subscript e and denote the second piola kirchhoff effective stress tensor of the solid. In this section we formulate the balance of energy or the first law of. However, several other measures of stress can be defined.
The 1st piolakirchhoff stress is energy conjugate to the deformation gradient. Now the second piolakirchhoff stress tensor with respect to the principal axes of the tissue, s. The 1st piola kirchhoff stress is the 3d generalization of the 1d concept of engineering stress. Cauchy stress tensor kirchhoff stress tensor first piola kirchhoff stress tensor ylhis paper is presented to my good friend, prof. Stress and strain are called 2nd order tensor because they follow the transformation rule. The stress tensor the general state of stress in any homogeneous continuum, whether. The stress tensor is a representation of stress on three mutually perpendicular planes in a coordination system. Stress balance principles 03 the cauchy stress tensor. On the irredundant part of the first piolakirchhoff stress. Combining this with 2, and taking into account that trh trhnew 0, the condition 68 is. The second piola kirchoff stress tensor is the derivative of the helmholtz free energy with respect to the green strain tensor m. The first piola kirchhoff is an unsymmetric tensor, and is for that reason less attractive to work with.
Is it just outputting stress and strain of one element and combine them together. Optimal control problems in finitestrain elasticity by. Since we were operating in the realm of small deformation mechanics, then the cauchy stress tensor is equivalent to the other stress tensors. Sinceaj isnotspecifiedwithrespect toafixedconfiguration. Note that stress tensor is symmetrical, there are 6 independent variables instead of 9.
But in the documentation it is also spoken about 2nd piola kirchhoff and about large and small deformations, but not how to read it from the results file. Sometimes you may also encounter the kirchhoff stress. Indeed, following, the strain energy density function w is 44 w c 1 2. Coordinate transformations of 2nd rank tensors were discussed on this coordinate transform page. It can also be shown that it is energitically correct. We calculate the viscous contribution to the total stress state in the lagrangian configuration using the second piola kirchhoff effective stress tensor s e s j s f s. The second piolakirchhoff stress tensor can be computed from w as s wewhile the. The static spherically symmetric body in relativistic. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. N ds df i ij j nn piola kirchhoff 1 stress tensor pk1. To recall, it is the limiting value of the ratio of force over area.1134 349 742 1371 830 314 465 547 577 652 688 936 614 1135 1235 1645 501 1025 155 874 932 1106 55 1002 397 481 538 412 1105 1098 883 402 1276 257 1346 106 356